Semiconductor devices such as logic and memory devices are typically fabricated by a sequence of processing steps applied to a specimen. The various features and multiple structural levels of the semiconductor devices are formed by these processing steps. For example, lithography among others is one semiconductor fabrication process that involves generating a pattern on a semiconductor wafer. Additional examples of semiconductor fabrication processes include, but are not limited to, chemical-mechanical polishing, etch, deposition, and ion implantation. Multiple semiconductor devices may be fabricated on a single semiconductor wafer and then separated into individual semiconductor devices.
Optical metrology processes are used at various steps during a semiconductor manufacturing process to detect defects on wafers to promote higher yield. Optical metrology techniques offer the potential for high throughput without the risk of sample destruction. A number of optical metrology based techniques including scatterometry and reflectometry implementations and associated analysis algorithms are commonly used to characterize critical dimensions, overlay, film thicknesses, process parameters, composition and other parameters of nanoscale structures.
As devices (e.g., logic and memory devices) move toward smaller nanometer-scale dimensions, characterization becomes more difficult. Devices incorporating complex three-dimensional geometry and materials with diverse physical properties contribute to characterization difficulty.
In response to these challenges, more complex optical tools have been developed. Measurements are performed over a large ranges of several machine parameters (e.g., wavelength, azimuth and angle of incidence, etc.), and often simultaneously. As a result, the measurement time, computation time, and the overall time to generate reliable results, including measurement recipes, increases significantly.
Existing model based metrology methods typically include a series of steps to model and then measure structure parameters. Typically, measurement data (e.g., measured data, DOE data, etc.) is collected from a particular metrology target. An accurate measurement model of the optical system, dispersion parameters, and geometric features is formulated. An electromagnetic (EM) solver is employed to solve the measurement model and predict measurement results. A series of simulations, analysis, and regressions are performed to refine the measurement model and determine which model parameters to float. In some examples, a library of synthetic spectra is generated. Finally, measurements are performed using the library or regression in real time with the measurement model.
The EM simulation process is controlled by a number of parameters (e.g., slabbing parameters, Rigorous Coupled Wave Analysis (RCWA) parameters, discretization parameters, etc.). Simulation parameters are selected to avoid introducing excessively large errors. However, in general, there is a trade-off between computational effort and solution accuracy. In other words, an accurate solution requires much more computational effort than a less accurate solution. Currently, the computational effort required to arrive at sufficiently accurate measurement results for complex semiconductor structures is large and growing larger.
Many EM simulation algorithms are based on spatial Fourier harmonic expansions of the dielectric permittivity of a target and of the electric and magnetic fields incident and scattered by the target. These algorithms are widely used in semiconductor metrology due to their stability, and ability to achieve the desired accuracy with relatively high speed. Exemplary algorithms include Rigorous Coupled Wave Analysis (RCWA), Classical Modal Method, Finite Difference methods, etc. These algorithms are typically employed to compute electromagnetic scattering by periodic targets. The algorithms use Fourier expansions of the periodic targets and the electromagnetic fields in terms of spatial harmonics. In principle, Fourier series expansions have an infinite number of terms. However, in practical computations by digital computers, a truncated version of the Fourier series expansion having a finite number of Fourier harmonics in a range between a minimum and a maximum spatial frequency are employed. The truncation order (TO) of the Fourier series expansion is commonly identified as the highest order spatial harmonic of the truncated Fourier expansion.
Many current metrology systems employ a RCWA algorithm as the EM simulation engine employed to solve the measurement model. Simulated measurement signals are computed by the RCWA engine. In some embodiments, measured signals are compared to the computed signals as part of a regression analysis to estimate measurement parameter values.
To simulate measurement signals generated by a periodic metrology target using RCWA, the profiles of periodic structures are approximated by a number of sufficiently thin planar grating slabs. RCWA involves three main steps: 1) Fourier expansion of the electric and magnetic fields inside the grating, 2) Solution of Maxwell's equations by calculation of the eigenvalues and eigenvectors of a constant coefficient matrix that characterizes the diffracted signal, or an equivalent method, and 3) Solution of a linear system deduced from the boundary matching conditions. The analysis is divided into three distinct spatial regions: 1) the ambient region supporting the incident plane wave field and a summation over all reflected diffracted orders, 2) the grating structure and underlying non-patterned layers where the wave field is treated as a superposition of modes associated with each diffracted order, and 3) the substrate containing the transmitted wave field.
The accuracy of the RCWA solution depends, in part, on the number of terms retained in the space-harmonic expansion of the wave fields. The number of terms retained is a function of the number of spatial harmonic orders considered during the calculations. Efficient generation of a simulated diffraction signal for a given hypothetical profile involves selection of the optimal set of spatial harmonics orders at each wavelength for transverse-magnetic (TM) components of the diffraction signal, transverse-electric (TE) components of the diffraction signal, or both. Mathematically, the more spatial harmonic orders selected, the more accurate the simulations. However, this comes at a price of higher computational effort and memory consumption. Moreover, the computational effort and memory consumption is a strongly nonlinear function of the number of orders used. Typically, computational effort scales with the third power for simulations of two dimensional structures and scales with the sixth power for three dimensional structures. Similarly, memory consumption scales with the second power for two dimensional structures and to the fourth power for three dimensional structures.
The importance of selecting the appropriate number of spatial harmonic orders increases significantly when three-dimensional structures are considered in comparison to two-dimensional structures. Since the selection of the number of spatial harmonic orders is application specific, efficient approaches for selecting the number of spatial harmonics orders can be critical to achieve sufficiently accurate results in a reasonable period of time.
In some examples, a compact pattern of spatial harmonics is selected. In these examples a single truncation order (TO) is selected in each direction of periodicity of the target, and all Fourier harmonics within the range of the selected TO are typically used. For example, if the target is periodic in one direction (e.g., a two dimensional line-space grating, etc.), a single TO is determined by trading off computation time for simulation accuracy, and all of the spatial harmonics in the range {−TO, +TO} are employed. If the target is periodic in two directions (e.g., an array of contact holes, two crossed-gratings, etc.), then a TO associated with each direction (e.g., TOx and TOy) is selected in a similar manner. Similarly, all of the spatial harmonics in the rectangular region with corners (−TOx, −TOy), (+TOx, −TOy), (+TOx, +TOy), and (−TOx, +TOy) are employed in the simulation.
In some examples, a sparse pattern of spatial harmonics is selected. U.S. Patent Publication No. 2011/0288822 A1 by Veldman et al. and U.S. Pat. No. 7,428,060 B2 to Jin et al., incorporated herein by reference in their entirety, describe the selection of non-rectangular patterns of Fourier modes for three dimensional grating structures based on the convergence of the computation algorithm.
However, these approaches to selecting the pattern of spatial harmonics become problematic when the periodic structure has two or more characteristic repeating length scales in one or more directions of periodicity, particularly when one or more of the repeating length scales is relatively large. In these approaches, a large period requires a large truncation order, even if the other repeating length scales are relatively small. Hence, in these approaches, the TO is dictated by the largest pitch.
When current systems are employed to measure complex geometric structures, three dimensional structures, and structures having multiple periods in each direction, a high truncation order is necessary to accurately represent the corresponding physical measurement signals. This significantly increases the required computational effort. In some examples, when faced with multiple pediodicity, EM simulation algorithms commonly used in metrology can be slowed by several orders of magnitude relative to single-period structures.
To meet the increasing computational burden, large computing clusters are required, and in some cases it is impractical to perform the necessary computations for some models. Although a lower truncation order may be employed to reduce the required computational effort, this often results in unacceptably large measurement errors.
Increasingly complicated measurement models are causing corresponding increases in computational effort. Improved model solution methods and tools are desired to arrive at sufficiently accurate measurement results with reduced computational effort.